Concepts from Chemical Reaction Kinetics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 230

The above formalism also allows the elucidation of reaction rates when more than
one species is involved in the reaction. The Langmuir isotherm for two competing
species A and B gives

K Lang,A P A

θ A = ,

1 + K Lang,A P A + K Lang,B P B

(5.137)

K Lang,B P B

θ B = .

1 + K Lang,A P A + K Lang,B P B

If the heterogeneous catalysis is bimolecular involving both adsorbed species A and
B (Langmuir-Hinshelwood mechanism), we have the following overall rate:

K Lang,A K Lang,B P A P B

r = k θ A θ B = k

K Lang,B P B ) 2 .

(5.138)

( 1 + K Lang,A P A +

Note that the above equation indicates the competition for surface sites between A
and B. As a consequence, if P A is held constant, the rate will go through a maximum
as P B is varied.
If the heterogeneous catalysis is bimolecular involving adsorbed B reacting with
the gas-phase species A (Langmuir-Rideal mechanism), we have the following rate:

K Lang,B P A P B
( 1 + K Lang,A P A + K Lang,B P B ) .

r = k θ B P A = k (5.139)

By replacing pressure P A with the aqueous concentration [A], we obtain the rate of
heterogeneous surface catalysis in solutions.
Let us now consider the energetics of heterogeneous catalysis. Consider the rate
of the unimolecular reaction at low pressures:

k P A , r = kK Lang,A P A = (5.140)

where k is the overall first-order rate constant. Note that the first-order rate constant
k varies with T according to the Arrhenius expression

dln k
d T =

E a
RT 2 .

(5.141)

Similarly, as discussed in Section 3.5, the Langmuir adsorption constant K Lang,A also
has a relationship with T ,

dln K Lang,A
d T

q ads
RT 2 ,

=− (5.142)

where q ads is the heat evolved during adsorption. Hence
dln k
d T =

E a

(E a − q ads )
RT 2

=

RT 2 .

(5.143)

The true activation energy E a is smaller than E a by the quantity q ads as shown
schematically in Figure 5.13. For the case of high pressures, E a is the same as E a .